reflectivity & amorphous

Reflectivity & Amorphous

Amorphous Materials

Periodic arrangement of atoms causes destructive interference in all directions except those predicted by Bragg’s law.

The measurable diffraction occurs at non-Bragg angles only when crystal imperfections are present.

Amorphous materials do not show long-range order. They exhibit short-range order: statistical preference of a particular interatomic distance.

Amorphous Materials

Consider our sample as any form of matter in which there is random orientation.

This includes gases, liquids, amorphous solids, and crystalline powders.

The scattered intensity from such sample:

rmnf

s – s0

s0 s

n

i

nm

mn

i

nm

m

n

i

n

m

i

m

mnmn

nm

effeffI

efefI

rqrss

rssrss

0

00

2

22,

where

nmmn rrr takes allorientations

)(2

0ssq

Amorphous Materials

Average intensity from an array of atoms which takes all orientations in space:

rmnf

s – s0

s0 s

,sin

n mn

mnnm

m qr

qrffI where

Debye scattering equation

sin4q

It involves only the magnitudes of the distances rmn of each atom from every other atom

Bragg’s law

Polyatomic Molecules

Consider gas of polyatomic molecules.

Gas is not too dense – there is complete incoherency between the scattering by different molecules.

Intensity per molecule:

Lets take a carbon tetrachloride as example.

It is composed of tetrahedral molecules CCl4.

Then:

Rqr

qrffNI

n mn

mnnm

m

sin correction

factor

ClCqr

ClCqrf

ClCqr

ClCqrfff

ClCqr

ClCqrfffNI

ClCClCl

ClCC

sin3

sin4

sin4

CCl4

Polyatomic Molecules

For tetrahedral CCl4 molecule:

The intensity depends on just one distance r(C – Cl).

Peaks and dips do not require the existence of a crystalline structure.

Certain interatomic distances that are more probable than others are enough to get peaks and dips on the scattering curve.

ClCrClClr 3

8

Intensity (in e.u. per molecule) for a CCl4 gas in which the C – Cl distance is r = 1.82 Å. (Warren)

RClCqr

ClCqrf

ClCqr

ClCqrffffNI

Cl

ClCClC

sin12

sin84

2

22

Å82.1ClCr

Crystal as Molecule

Lets treat crystal as a molecule

FCC has 14 atoms:

8 at corners of a cube

6 at face center positions

a is the edge of a cube

Rqr

qrffNI

n mn

mnnm

m

sin

3

3sin8

2

2sin24

5.1

5.1sin48

sin30

2

2sin72142

qa

qa

qa

qa

qa

qa

qa

qa

aq

aqNfI

qa

Amorphous Materials

Atoms in liquids and amorphous solids have definite structures relative to an origin at the center of an average atom.

This type of structure is expressed by a radial distribution function:

We use the equation:

It can be shown that the scattered intensity can be written as:

rr 24

drrr 24 – average number of atom centers between distances r and r + dr from the center of an average atom

n mn

mnnm

m qr

qrffI

sin

dzzAedzzAefdrqr

qrrrNfNfI ikzikz2

0

2

00

222 sin4

interaction between distant neighbors

interaction between near neighbors not negligible only for

very small angles

Assumption: Sample takes with equal probability all orientations in space

Amorphous Materials

We obtain very important and much used equation:

For solids:

4r 2(r )dr – average number of atom centers between distances r and r + drfrom the center of an average atom.

For liquids:

4r 2(r )dr – average over surroundings of each atom in the sample and also an average over the time of measurement.

0 20

22 sin12

44 dqrqNf

Iq

rrrr

0 – average atom density per sample can be obtained experimentally from the scattering curve

Experimental determination of 4r 2(r )

Previous expressions apply only to coherent (unmodified) scattering.

Many corrections are required:

correct for air scatter

correct for absorption by sample

correct for polarization

correct for incoherent (Compton-modified) scattering – requires conversion to absolute (electron) units

RDF determination requires high-quality data at large q (small r ).

Example: Liquid Sodium

We obtain :

(a) Total intensity curve for liquid Na unmodified + modified, (b) total independent scattering per atom, (c) independet unmodified scattering per atom f 2, (d) modified scattering per atom i (M ). (Warren)

c

ba

c

cda

Nf

fI

Nf

Iqi

2

2

21

12

Nf

Iqi

Example: Liquid Sodium

Experimental curve qi(q) for liquid Na. (Warren) (a) RDF 4r 2(r ) for liquid Na, (b) average density curve 4r 20(r ), (c) distribution of neighbors in crystalline Na. (Warren)

0

0

22 sin2

44 dqrqqiqr

rrr

Example: Carbon black

(a) Final scattering curve of carbon black, theoretical independent scattering curves: (d) coherent, (c) incoherent, and (b) total independent scattering.

What is Carbon Black?

Carbon black is made primarily from a petroleum-based

feedstock. The oil is pumped into a specially designed

furnace, where it is heated above 2,000° F. This process

"cracks" the oil to produce a gas stream laden with carbon

black powder. The gas stream passes through a series of

filters, where the carbon black is separated from the gases.

The carbon black powder then is bound with water to create

larger beads or granules, which are passed through a dryer

and packaged for delivery to customers in every part of the

world.

Actual measurement

Example: Carbon black

Plot of the experimental amplitude function S i(S) for carbon black

RDF of carbon black

X-ray Reflectivity

X-ray reflectivity is a precise and non-destructive method used to determine the layer thickness, density and roughness of a layer on a substrate.

11

2

refracted beam

incident beamreflected beam

n1 – air

n2 – sample

Sample

Air

X-ray Reflectivity

An X-ray-beam that strikes a solid-surface at a small angle (0-2°) is totally reflected.

Above the critical angle of total reflectance c the beam penetrates the sample, whereby the angle of refraction 2 is smaller than the angle of incidence 1 (the refractive index of X-rays in solids is smaller than that in air).

The refractive index for X-ray radiation is given by the formula:

According to Snell´s law of refraction:

1nn – refractive index, – term, that specifies the dispersion of the x-ray beam.

1

2

122211 coscoscoscos

n

nnn

if n1 – index of refraction of air ~ 1n2 – index of refraction in solid < 1 then 12

X-ray Reflectivity

Below critical angle total reflection occurs:

The density of the sample can be calculated from the critical angle and using the following equation:

j

jj

j

jA fZA

rN

2

2

02

222

12221

2

221arccos

11cos0coscos

0

C

CC nnnn o

o

since

then

NA – Avogadro-Numberr0 – classical radius of an electron – wavelengthj – density of the atom j in the compoundAj – atomic mass of the atom jZj – atomic number of the atom jf‘j – correction factor for the dispersion for the atom j

Incident angle (deg)

21.91.81.71.61.51.41.31.21.110.90.80.70.60.50.40.30.20.10

Inten

sity (c

ounts

/s)

0

1

10

100

1,000

10,000

100,000

1,000,000

X-ray Reflectivity

Reflectivity curve for Si

critical angle

beamprimarytheofIntensity

beamreflectedtheofIntensityR tyReflectivi

X-ray Reflectivity from Thin Layers

If the sample contains a thin layer, x-rays are reflected from the air/layer as well as from the layer/substrate interfaces.

11

2

incident beamreflected beam

n1 – air

n2 – layer

Samplen3 – substrate

t

A

B C D

path difference: BCD


21.91.81.71.61.51.41.31.21.110.90.80.70.60.50.40.30.20.10

Inte

nsity

(cou

nts/

s)

0

1

10

100

1,000

10,000

100,000

1,000,000


Positions of the maxima can be calculated using Bragg’s law:

222 sin2sinsin tttCDBCnL

SiO2/Si

B DC

1

2t


Some simplifications and approximations:

2sintCDBC

B DC

1

2t

if is small sin

using Snell’s law of refraction (n1 ~ 1):

221221 cos1coscoscos n

2

2

1

2

12

2

12 2

1

cosarccos

1

coscos

2

2

12 2

2

2

12 22sin2 ttnL

2

2

2

2

1 24

nt

y = a x + b


0.50.40.30.2

Intensi

ty (cou

nts/s)

5,000

50,000

500,000

Example: Si on Ta

2

2

2

2

1 24

nt

y = a x + b

4 6 8 10 12 14 16 18 20

Example: Si on Ta

0 4 6 8 10 12 14 16 18 20

0.1

0.2

0.3

0.4

0.5

n

1

(de

g)

2

2

2

2

1 24

nt

y = a x + b

a

b

a

b

2

4

222

2

0

2

2

2

02

t

fZrN

A

t

fZA

rN

A

A

NA – Avogadro-Numberr0 – classical radius of an electron – wavelengthj – density of the atom j in the compoundAj – atomic mass of the atom jZj – atomic number of the atom jf ‘j – correction factor for the dispersion for the atom j

We get:

t = 181 nm = 2.2 g/cm3


Vladimir Kogan, PANalytical


3.83.63.43.232.82.62.42.221.81.61.41.210.80.60.40.2

Inten

sity (

coun

ts/s)

0

1

10

100

1,000

10,000

100,000

X-ray Reflectivity: Simulation

Reflectivity can be calculated completely using Fresnell equations

2

4

21

4

21

22

22

1

ti

ti

eRR

eRRR

Simple version for substrate + one layer

R1 and R2 reflectivities of air/layer and layer/substrate interfaces:

2

221

24

21

21

eR

– surface roughness

PbTiO3/SrTiO3


3.83.63.43.232.82.62.42.221.81.61.41.210.80.60.40.2

Inten

sity (c

ounts

/s)

0

1

10

100

1,000

10,000

100,000

X-ray Reflectivity: Roughness

PbTiO3 on SrTiO3

Calculation with roughness = 0

Measured Reflectivity


Smooth surface Rough surface

http://annwm.lbl.gov/pictures/yosemite_05-13-06/thumb/viewF.cgi?IMG_1111e.JPG

http://annwm.lbl.gov/pictures/yosemite_05-13-06/thumb/viewF.cgi?IMG_1111e.JPG


Reflectivity of water

Braslau et al. Phys. Rev. Lett. 54 114 (1985)

Fresnel Reflectivity


Difference between experiment and theory due to roughness


3.83.63.43.232.82.62.42.221.81.61.41.210.80.60.40.2

Intensi

ty (cou

nts/s)

0

1

10

100

1,000

10,000

100,000

X-ray Reflectivity: Density

PbTiO3 on SrTiO3

Overestimated density


X-ray Reflectivity: Density

PbTiO3 on SrTiO3

Densities are similar


3.83.63.43.232.82.62.42.221.81.61.41.210.80.60.40.2

Inten

sity (c

ounts

/s)

0

1

10

100

1,000

10,000

100,000

X-ray Reflectivity: Thickness

PbTiO3 on SrTiO3

Overestimated thickness


X-ray Reflectivity: Fitting

PbTiO3 on SrTiO3


3.83.63.43.232.82.62.42.221.81.61.41.210.80.60.40.2

Intensi

ty (cou

nts/s)

0

1

10

100

1,000

10,000

100,000

X-ray Reflectivity: Simulation

Self-Assembled Monolayers C18H37SH on Au

Vladimir Kogan, PANalytical

Specular Reflectivity Curve Reflectivity Map, Diffuse Scattering

Determined thickness of the layers:C18H37SH - 1.6nmAu1 - 0.6nmAu2 - 19.0nmSi > 100,000nm

Determined:Average Lateral Correlation Length: 2.5nm

X-ray Reflectivity

10 X (D1 1nm + D2 1nm)10 X (D1 2nm + D2 2nm)

10 X (D1 10nm + D2 10nm) 25 X (D1 2nm + D2 2nm)

X-ray Reflectivity25 X (D1 2nm + D2 2nm)

25 X (D1 2nm + D2 2nm)+ Roughness

1 - D1 2nm

2 - D2 5nm

3 - D1 3nm

4 - D2 15nm

5 - D1 25nm

6 - D2 7nm

7 - D1 14nm

8 - D2 6nm

9 - D1 3nm

10 - D2 11nm

11 - D1 14nm

12 - D2 5nm

Multilayer

X-ray Reflectivity

Small Angle Scattering

Using 2D detector

Epitaxial Layer

Structure

Epitaxial Layer

Structure

Thin Filmsepitaxial

polycrystallineamorphous

Rocking Curve

Analysiswith high resolution

optics

Reciprocal Space Mapsusing triple-axis analyzer

Reflectometry and thin film

phase analysiscomposition, layer

thickness and interface quality

SchematicSample

Schematic Beam Path Applications Example

In-plane diffractionfrom very thin films.

Depth sensitivity

Phase analysis and

Omega-stress

with Bragg-Brentano geometry

Phase analysis and

Omega-stress

with parallel beam optics

Very thin films

Polycrystalline material. Flat

surface

Polycrystalline material.

Rough surface

SchematicSample


Very small sample. Spot on a sample

Solid sample

Psi-stress and texture

analysisusing point focus with lens and

parallel plate collimator

optics

Spot analysis on small and

inhomogeneous samples

using mono-capillary optics

SchematicSample


reflectivity & amorphous

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